I love platformers, both 2D and 3D. (Super Mario 64 and Sands of Time are my favorites.)
Also a big fan of Zelda games. Wind Waker tops the list, but Majora's Mask has some mechanics that I adore. (Time travel and the Bomber's Notebook? Heck yes!)
I hate RPGs. Stats, items and turn-based battles... Yuck. Also not big on strategy games, but that's mostly because I suck at them. Advance Wars is cool, though.

It's a toss up between RPG (say, Fallout 3) and strategy games (Total War series). I'm really no fan of micromanaging clickfests that provide a challenge by letting the computer build way more units more rapidly than the player can.